Global classical solvability and stabilization in a two-dimensional chemotaxis–fluid system with sub-logarithmic sensitivity

In this paper, we consider the following system: [Formula: see text] in a smoothly bounded domain [Formula: see text] with [Formula: see text] and a given function [Formula: see text] with [Formula: see text] It is proved that if [Formula: see text] then for appropriately small initial data an associated no-flux/no-flux/Dirichlet initial-boundary value problem is globally solvable in the classical sense, and that if [Formula: see text] then under a different but still suitable smallness restriction of the initial data, a corresponding initial-boundary value problem subject to no-flux/no-flux/Dirichlet boundary conditions admits a unique classical solution which is globally bounded and approaches a constant equilibria [Formula: see text] in [Formula: see text] as [Formula: see text] with [Formula: see text]